Thermodynamic Invariants of Coupled Channels: A Many-Channel Tolman-Ehrenfest Effect

When multiple thermodynamic channels are coupled, single-channel equilibrium conditions fail. Extending the Tolman–Ehrenfest effect to the entropy manifold, we derive the unique $n$-channel invariant ${\zeta }_i{T}_i=C$, where ${\zeta }_i$ is the holonomy of the Ruppeiner connection. For the granular volume–stress ensemble, Rowe’s dilatancy ratio and energy restriction emerge as geometric consequences of the off-diagonal curvature $g_{V\sigma }$, and the 60-year puzzle of state-dependent $K_{\mu }$ is resolved: the correction ${\zeta }_V$ reaches $O(1)$ near jamming. The prediction ${\zeta }_V\chi =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ across a shear band is experimentally testable.